The aim of the information bottleneck is to compress the information in \({\bf y}\)  into \({\bf t}\)  following the rate constraint \(R\) , such that \({\bf t}\)  can provide the most useful information about \({\bf x}\)  in the mutual information sense.
In this context, it has been recognized that information bottleneck problem, provides a method to extract the information of \({\bf y}\)  which is most relevant for estimating \({\bf x}\)  [1]} or to extract the information in \({\bf y}\)  which is most relevant in approximating \({\bf x}\) .
Now let us consider a toy example of sequential decision making for two agents, where one receives an observation \({\bf y}\)  that has to be, at least partly, communicated to the other one in order to maximize the expected cumulative reward. The random variable \({\bf x}\)  can be seen as the expected cumulative reward of the system given action \(a\)  and the current state of the system, where the state is jointly defined by \(\langle {\bf y}, {\bf y}^{^{\prime }} \rangle \) . The parameter \({\bf t}\)  is the communication message that the first agent is about to transmit based on its observation \({\bf y}\)  to facilitate the control decision \(a\)  by the agent \(j\) . Accordingly, by solving a (conditional) information bottleneck problem, at the side of encoder, we can optimize the communication of observations \({\bf y}\)  of agent \(i\) , by compressing them to \({\bf t}\)  while ensuring that the compressed communication message \({\bf t}\)  has yet the maximum possible information about the conditional expected return \({\bf x}\) . Fig. REF  illustrates the setting of this problem. When switch (1) and (2) are both off the problem reduces to a rate-distortion problem with memory at encoder of the agent \(i\) . The work done in [2]} was first to show the connection of the information bottleneck problem and rate-distortion with logarithmic loss. To stay consistent with the framework that will be proposed in section , we assume the switch (1) to be always on. Accordingly, conditioned on the switch (2) being on, the coding in the encoder of agent \(j\)  should be done such that it maximizes the conditional mutual information \(I\big ( {\bf x}; {\bf t}| {\bf y}^{^{\prime }} \big )\) . A solution to the conditional information bottleneck problem is provided by [3]}.
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